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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.students_t_dist"></a><a class="link" href="students_t_dist.html" title="Students t Distribution">Students
        t Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">students_t_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">students_t_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">students_t</span><span class="special">;</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">students_t_distribution</span>
<span class="special">{</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>

   <span class="comment">// Constructor:</span>
   <span class="identifier">students_t_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>

   <span class="comment">// Accessor:</span>
   <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>

   <span class="comment">// degrees of freedom estimation:</span>
   <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
      <span class="identifier">RealType</span> <span class="identifier">difference_from_mean</span><span class="special">,</span>
      <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
      <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span>
      <span class="identifier">RealType</span> <span class="identifier">sd</span><span class="special">,</span>
      <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span>
<span class="special">};</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
          Student's t-distribution is a statistical distribution published by William
          Gosset in 1908. His employer, Guinness Breweries, required him to publish
          under a pseudonym (possibly to hide that they were using statistics to
          improve beer quality), so he chose "Student".
        </p>
<p>
          Given N independent measurements, let
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/students_t_dist.svg"></span>

          </p></blockquote></div>
<p>
          where <span class="emphasis"><em>M</em></span> is the population mean, μ is the sample mean,
          and <span class="emphasis"><em>s</em></span> is the sample variance.
        </p>
<p>
          <a href="https://en.wikipedia.org/wiki/Student%27s_t-distribution" target="_top">Student's
          t-distribution</a> is defined as the distribution of the random variable
          t which is - very loosely - the "best" that we can do while not
          knowing the true standard deviation of the sample. It has the PDF:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/students_t_ref1.svg"></span>

          </p></blockquote></div>
<p>
          The Student's t-distribution takes a single parameter: the number of degrees
          of freedom of the sample. When the degrees of freedom is <span class="emphasis"><em>one</em></span>
          then this distribution is the same as the Cauchy-distribution. As the number
          of degrees of freedom tends towards infinity, then this distribution approaches
          the normal-distribution. The following graph illustrates how the PDF varies
          with the degrees of freedom ν:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/students_t_pdf.svg" align="middle"></span>

          </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.students_t_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.member_functions"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">students_t_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
</pre>
<p>
          Constructs a Student's t-distribution with <span class="emphasis"><em>v</em></span> degrees
          of freedom.
        </p>
<p>
          Requires <span class="emphasis"><em>v</em></span> &gt; 0, including infinity (if RealType
          permits), otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
          Note that non-integral degrees of freedom are supported, and are meaningful
          under certain circumstances.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          returns the number of degrees of freedom of this distribution.
        </p>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
   <span class="identifier">RealType</span> <span class="identifier">difference_from_mean</span><span class="special">,</span>
   <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
   <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span>
   <span class="identifier">RealType</span> <span class="identifier">sd</span><span class="special">,</span>
   <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span>
</pre>
<p>
          returns the number of degrees of freedom required to observe a significant
          result in the Student's t test when the mean differs from the "true"
          mean by <span class="emphasis"><em>difference_from_mean</em></span>.
        </p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">difference_from_mean</span></dt>
<dd><p>
                The difference between the true mean and the sample mean that we
                wish to show is significant.
              </p></dd>
<dt><span class="term">alpha</span></dt>
<dd><p>
                The maximum acceptable probability of rejecting the null hypothesis
                when it is in fact true.
              </p></dd>
<dt><span class="term">beta</span></dt>
<dd><p>
                The maximum acceptable probability of failing to reject the null
                hypothesis when it is in fact false.
              </p></dd>
<dt><span class="term">sd</span></dt>
<dd><p>
                The sample standard deviation.
              </p></dd>
<dt><span class="term">hint</span></dt>
<dd><p>
                A hint for the location to start looking for the result, a good choice
                for this would be the sample size of a previous borderline Student's
                t test.
              </p></dd>
</dl>
</div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
            Remember that for a two-sided test, you must divide alpha by two before
            calling this function.
          </p></td></tr>
</table></div>
<p>
          For more information on this function see the <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm" target="_top">NIST
          Engineering Statistics Handbook</a>.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.students_t_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.non_member_accessors"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
          functions</a> that are generic to all distributions are supported:
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          The domain of the random variable is [-∞, +∞].
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.students_t_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.examples"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.examples">Examples</a>
        </h5>
<p>
          Various <a class="link" href="../../stat_tut/weg/st_eg.html" title="Student's t Distribution Examples">worked examples</a>
          are available illustrating the use of the Student's t distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.students_t_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.accuracy"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The normal distribution is implemented in terms of the <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete
          beta function</a> and <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">its
          inverses</a>, refer to accuracy data on those functions for more information.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.students_t_dist.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.implementation0"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.implementation0">Implementation</a>
        </h5>
<p>
          In the following table <span class="emphasis"><em>v</em></span> is the degrees of freedom
          of the distribution, <span class="emphasis"><em>t</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
          is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
        </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Function
                  </p>
                </th>
<th>
                  <p>
                    Implementation Notes
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    pdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: <span class="serif_italic">pdf = (v / (v
                    + t<sup>2</sup>))<sup>(1+v)/2 </sup> / (sqrt(v) * <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>(v/2,
                    0.5))</span>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relations:
                  </p>
                  <p>
                    <span class="serif_italic">p = 1 - z <span class="emphasis"><em>iff t &gt; 0</em></span></span>
                  </p>
                  <p>
                    <span class="serif_italic">p = z <span class="emphasis"><em>otherwise</em></span></span>
                  </p>
                  <p>
                    where z is given by:
                  </p>
                  <p>
                    <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v
                    / 2, 0.5, v / (v + t<sup>2</sup>)) / 2 <span class="emphasis"><em>iff v &lt; 2t<sup>2</sup></em></span>
                  </p>
                  <p>
                    <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(0.5,
                    v / 2, t<sup>2 </sup> / (v + t<sup>2</sup>) / 2 <span class="emphasis"><em>otherwise</em></span>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: q = cdf(-t)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: <span class="serif_italic">t = sign(p -
                    0.5) * sqrt(v * y / x)</span>
                  </p>
                  <p>
                    where:
                  </p>
                  <p>
                    <span class="serif_italic">x = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v
                    / 2, 0.5, 2 * min(p, q)) </span>
                  </p>
                  <p>
                    <span class="serif_italic">y = 1 - x</span>
                  </p>
                  <p>
                    The quantities <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>y</em></span>
                    are both returned by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>
                    without the subtraction implied above.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: t = -quantile(q)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    0
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    0
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    variance
                  </p>
                </td>
<td>
                  <p>
                    if (v &gt; 2) v / (v - 2) else NaN
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    if (v &gt; 3) 0 else NaN
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    if (v &gt; 4) 3 * (v - 2) / (v - 4) else NaN
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    if (v &gt; 4) 6 / (df - 4) else NaN
                  </p>
                </td>
</tr>
</tbody>
</table></div>
<p>
          If the moment index <span class="emphasis"><em>k</em></span> is less than <span class="emphasis"><em>v</em></span>,
          then the moment is undefined. Evaluating the moment will throw a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
          unless ignored by a policy, when it will return <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">();</span></code>
        </p>
<h6>
<a name="math_toolkit.dist_ref.dists.students_t_dist.h5"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.implementation"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.implementation">Implementation</a>
        </h6>
<p>
          (By popular demand, we now support infinite argument and random deviate.
          But we have not implemented the return of infinity as suggested by <a href="http://en.wikipedia.org/wiki/Student%27s_t-distribution" target="_top">Wikipedia
          Student's t</a>, instead throwing a domain error or return NaN. See
          also <a href="https://svn.boost.org/trac/boost/ticket/7177" target="_top">https://svn.boost.org/trac/boost/ticket/7177</a>.)
        </p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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